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<h1 class="heading"><a href="MATH-2023-OPDE.html"><span class="title">MATH 2023: Ordinary and Partial Differential Equations</span></a></h1>
<p class="byline">Xiaoyi Chen and Wei Zhang</p>
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<a href="ch_first.html" data-scroll="ch_first" class="internal"><span class="codenumber">1</span> <span class="title">Introduction</span></a><ul>
<li><a href="sec_1-intro.html" data-scroll="sec_1-intro" class="internal">Classification of Differential Equations</a></li>
<li><a href="sec_2-intro.html" data-scroll="sec_2-intro" class="internal">Linear and Nonlinear Equation</a></li>
<li><a href="sec_3-intro.html" data-scroll="sec_3-intro" class="internal">Geometrical Aspect</a></li>
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<a href="ch_second.html" data-scroll="ch_second" class="internal"><span class="codenumber">2</span> <span class="title">First Order Ordinary Differential Equations</span></a><ul>
<li><a href="sec2_1.html" data-scroll="sec2_1" class="internal">Linear Equations</a></li>
<li><a href="sec2_2.html" data-scroll="sec2_2" class="internal">Further Discussion of Linear Equations (For reading only)</a></li>
<li><a href="sec2_3.html" data-scroll="sec2_3" class="internal">Separable Equations</a></li>
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<a href="ch_third.html" data-scroll="ch_third" class="internal"><span class="codenumber">3</span> <span class="title">third Order Linear Equations</span></a><ul>
<li><a href="sec3_1.html" data-scroll="sec3_1" class="internal">Homogeneous equations with constant coefficient</a></li>
<li><a href="sec3_2.html" data-scroll="sec3_2" class="internal">Fundamental Solutions of Linear Homogeneous Equations</a></li>
<li><a href="sec3_3.html" data-scroll="sec3_3" class="internal">Linear Independence and Wronskian</a></li>
<li><a href="sec3_4.html" data-scroll="sec3_4" class="internal">Complex roots of the characteristic equations</a></li>
<li><a href="sec3_5.html" data-scroll="sec3_5" class="internal">Repeated Roots: Reduction of Order</a></li>
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<li><a href="sec4_1.html" data-scroll="sec4_1" class="internal">General Theory of the <span class="process-math">\(n\)</span>-th Order Linear Equations</a></li>
<li><a href="sec4_2.html" data-scroll="sec4_2" class="internal">Homogeneous Equations with Constant Coefficients</a></li>
<li><a href="sec4_3.html" data-scroll="sec4_3" class="internal">The Method of Undetermined Coefficients</a></li>
<li><a href="sec4_4.html" data-scroll="sec4_4" class="internal">The Method of Variation of Parameters</a></li>
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<a href="ch_five.html" data-scroll="ch_five" class="internal"><span class="codenumber">5</span> <span class="title">Series Solutions of Second Order Linear Equations</span></a><ul>
<li><a href="sec5_1.html" data-scroll="sec5_1" class="internal">Brief Review on Power Series</a></li>
<li><a href="sec5_2.html" data-scroll="sec5_2" class="internal">Introduction</a></li>
<li><a href="sec5_3.html" data-scroll="sec5_3" class="internal">Series Solutions Near an Ordinary Point</a></li>
<li><a href="sec5_4.html" data-scroll="sec5_4" class="internal">Euler’s Equation</a></li>
<li><a href="sec5_5.html" data-scroll="sec5_5" class="internal">Series Solution near a Regular Singular Point</a></li>
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<a href="ch_six.html" data-scroll="ch_six" class="internal"><span class="codenumber">6</span> <span class="title">System of First Order Linear Equations</span></a><ul>
<li><a href="sec6_1.html" data-scroll="sec6_1" class="internal">Introduction <span class="process-math">\(\&amp;\)</span> Basic Theory</a></li>
<li><a href="sec6_2.html" data-scroll="sec6_2" class="internal">Homogeneous System with Constant Coefficients</a></li>
<li><a href="sec6_3.html" data-scroll="sec6_3" class="internal">Complex Eigenvalues</a></li>
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<li><a href="sec6_5.html" data-scroll="sec6_5" class="internal">Fundamental Matrices</a></li>
<li><a href="sec6_6.html" data-scroll="sec6_6" class="internal">Non-homogeneous linear systems</a></li>
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<a href="ch_seven.html" data-scroll="ch_seven" class="internal"><span class="codenumber">7</span> <span class="title">Partial Differential Equations</span></a><ul>
<li><a href="sec7_1.html" data-scroll="sec7_1" class="internal">Two-Point Boundary Value Problems</a></li>
<li><a href="sec7_2.html" data-scroll="sec7_2" class="internal">Eigenvalue Problems</a></li>
<li><a href="sec7_3.html" data-scroll="sec7_3" class="internal">Fourier Series</a></li>
<li><a href="sec7_4.html" data-scroll="sec7_4" class="internal">The Fourier Convergence Theorem</a></li>
<li><a href="sec7_5.html" data-scroll="sec7_5" class="internal">Even and Odd Functions</a></li>
<li><a href="sec7_6.html" data-scroll="sec7_6" class="internal">Introduction to Partial Differential Equations</a></li>
<li><a href="sec7_7.html" data-scroll="sec7_7" class="internal">1D Heat Equation; Solutions by Separation of Variable and Fourier Series</a></li>
<li><a href="sec7_8.html" data-scroll="sec7_8" class="internal">Other Heat Conduction Problems</a></li>
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<a href="ch_eight.html" data-scroll="ch_eight" class="internal"><span class="codenumber">8</span> <span class="title">Laplace transform</span></a><ul>
<li><a href="sec8_1.html" data-scroll="sec8_1" class="internal">What are Laplace Transforms, and Why?</a></li>
<li><a href="sec8_2.html" data-scroll="sec8_2" class="internal">Finding Laplace Transforms</a></li>
<li><a href="sec8_3.html" data-scroll="sec8_3" class="internal">Finding inverse transforms using partial fractions</a></li>
<li><a href="sec8_4.html" data-scroll="sec8_4" class="internal">Solving ODEs and ODE Systems</a></li>
<li><a href="sec8_5.html" data-scroll="sec8_5" class="internal">Step input and Impulse problems</a></li>
<li><a href="sec8_6.html" data-scroll="sec8_6" class="active">Laplace transform for PDE (heat equation)</a></li>
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<a href="ch_features.html" data-scroll="ch_features" class="internal"><span class="codenumber">9</span> <span class="title">Examples of PreTeXt features</span></a><ul><li><a href="sec_features-blocks.html" data-scroll="sec_features-blocks" class="internal">Environments and Blocks</a></li></ul>
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<li class="link"><a href="solutions-1.html" data-scroll="solutions-1" class="internal"><span class="codenumber">A</span> <span class="title">Selected Hints</span></a></li>
<li class="link"><a href="solutions-2.html" data-scroll="solutions-2" class="internal"><span class="codenumber">B</span> <span class="title">Selected Solutions</span></a></li>
<li class="link"><a href="appendix-1.html" data-scroll="appendix-1" class="internal"><span class="codenumber">C</span> <span class="title">List of Symbols</span></a></li>
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<main class="main"><div id="content" class="pretext-content"><section class="section" id="sec8_6"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">8.6</span> <span class="title">Laplace transform for PDE (heat equation)</span>
</h2>
<p id="p-501">In this section, we show how to use the Laplace transform to solve the one-dimensional heat equation. There are three main steps in order to solve a PDE using the Laplace transform:1. Begin by taking the Laplace transform with one of the two variables, usually <span class="process-math">\(t\text{.}\)</span> This will give an ODE of the transform of the unknown function.2. Solving the ODE, we shall obtain the transform of the unknown function.3. By taking the inverse Laplace transform, we obtain the solution to the original problem.</p>
<p id="p-502">Given a function <span class="process-math">\(u(x,t)\)</span> defined for all <span class="process-math">\(t &gt; 0\)</span> and assumed to be bounded we can apply the Laplace transform in <span class="process-math">\(t\)</span> considering <span class="process-math">\(x\)</span> as a parameter.</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[u(x, t)]=\int_0^{\infty} e^{-st} u(x, t) \mathrm{d} t=U(x, s).
\end{equation*}
</div>
<p class="continuation">In applications to PDEs we need the following:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
\begin{aligned}
{\mathcal L}[u_t(x, t)]&amp;=\int_0^{\infty} e^{-st} u_t(x, t) \mathrm{d} t=e^{-st} u(x, t)\big|_0^{\infty}+s \int_0^{\infty} e^{-st} u(x, t) \mathrm{d} t\\
&amp;=s U(x, s)-u(x, 0),
\end{aligned}
\end{equation*}
</div>
<p class="continuation">so we have</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[u_t(x, t)]=s U(x, s)-u(x, 0).
\end{equation*}
</div>
<p class="continuation">In exactly the same way we obtain</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[u_{tt}(x, t)]=s^2 U(x, s)-s u(x, 0)-u_t(x, 0).
\end{equation*}
</div>
<p class="continuation">We also need the corresponding transforms of the <span class="process-math">\(x\)</span> derivatives:</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[u_x(x, t)]=\int_0^{\infty} e^{-st} u_x(x, t) \mathrm{d} t=U_x(x, s),
\end{equation*}
</div>
<p class="continuation">and</p>
<div class="displaymath process-math" data-contains-math-knowls="">
\begin{equation*}
{\mathcal L}[u_{xx}(x, t)]=\int_0^{\infty} e^{-st} u_{xx}(x, t) \mathrm{d} t=U_{xx}(x, s).
\end{equation*}
</div>
<p id="p-503"><dfn class="terminology">Examples</dfn>1. Solve the following system</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq8_1.html">
\begin{equation*}
\begin{cases}
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\quad x&gt;0, ~t&gt;0,\\
u(x, 0^+)=0,\quad x&gt;0\\
u(0, t)=\delta(t),\quad \lim \limits_{x \to \infty} u(x, t)=0,\quad t&gt;0
\end{cases}
\end{equation*}
</div>
<p class="continuation">(a) We begin by taking the Laplace transform, with respect to <span class="process-math">\(t\text{,}\)</span> of both sides</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq8_1.html">
\begin{equation*}
{\mathcal L}[u_t(x, t)]=s {\mathcal L}[u(x, t)]={\mathcal L}[u_{xx}(x, t)].
\end{equation*}
</div>
<p class="continuation">Let <span class="process-math">\(L[u(x, t)]=U(x, s)\text{,}\)</span> then</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq8_1.html" id="Eq8_1">
\begin{equation}
s U=\frac{\mathrm{d}^2 U}{\mathrm{d} x^2} ~~\to~~ \frac{\mathrm{d}^2 U}{\mathrm{d} x^2} -s U=0.\tag{8.6.1}
\end{equation}
</div>
<p class="continuation">Notice that we have obtained an ODE for the unknown function <span class="process-math">\(U\text{.}\)</span>(b) The general solution of (<a href="" class="xref" data-knowl="./knowl/Eq8_1.html" title="Equation 8.6.1">(8.6.1)</a>) is</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq8_1.html">
\begin{equation*}
U(x, s)=A(s) e^{\sqrt{s} x}+B(s) e^{-\sqrt{s}x}
\end{equation*}
</div>
<p class="continuation">Apply the boundary conditions, with <span class="process-math">\({\mathcal L}[f(t)]=F(s)\text{,}\)</span> we obtain</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq8_1.html">
\begin{equation*}
U(0, s)={\mathcal L}[u(0, t)]={\mathcal L}(\delta(t))=1,
\end{equation*}
</div>
<p class="continuation">and</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq8_1.html">
\begin{equation*}
\lim \limits_{x \to \infty} U(x, s)=\lim \limits_{x \to \infty} \int_0^{\infty} e^{-st} u(x, t) \mathrm{d} t=\int_0^{\infty} e^{-st} \lim \limits_{x \to \infty} u(x, t) \mathrm{d} t=0.
\end{equation*}
</div>
<p class="continuation">The boundary condition</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq8_1.html">
\begin{equation*}
\lim \limits_{x \to \infty} U(x, s)=0 ~~\to~~ A(s)=0,
\end{equation*}
</div>
<p class="continuation">as for every fixed <span class="process-math">\(s&gt;0\text{,}\)</span> <span class="process-math">\(e^{\sqrt{s} x}\)</span> increases as <span class="process-math">\(x \to \infty\text{.}\)</span> Hence</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq8_1.html">
\begin{equation*}
U(0, s)=B(s)=1.
\end{equation*}
</div>
<p class="continuation">Therefore,</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq8_1.html">
\begin{equation*}
U(x, s)=e^{-\sqrt{s}x}.
\end{equation*}
</div>
<p class="continuation">(c) From the table of Laplace transform, we obtain the inverse Laplace transform as</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq8_1.html">
\begin{equation*}
{\mathcal L}^{-1}(e^{-\sqrt{s} x})=\frac{x}{2 \sqrt{ \pi t^3}} e^{-\frac{x^2}{4 t}}.
\end{equation*}
</div>
<p class="continuation">Hence</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq8_1.html">
\begin{equation*}
u(x, t)=\frac{x}{2 \sqrt{\pi t^3}} e^{-\frac{x^2}{4 t}}.
\end{equation*}
</div>
<p id="p-504">2. Solve the following system</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq9_1.html">
\begin{equation*}
\begin{cases}
\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},\quad 0&lt;x&lt;2, ~t&gt;0,\\
u(x, 0)=3 \sin (2 \pi x),\quad 0&lt;x&lt;2,\\
u(0, t)=0,\quad u(2, t)=0,\quad t&gt;0.
\end{cases}
\end{equation*}
</div>
<p class="continuation">(a) Take the Laplace transform and apply the initial condition</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq9_1.html">
\begin{equation*}
\frac{\mathrm{d}^2 U(x, s)}{\mathrm{d} x^2}=s U(x, s)-u(x, 0)=s U(x, s)-3 \sin (2 \pi x).
\end{equation*}
</div>
<p class="continuation">We write this equation as a non-homogeneous, second order linear constant coefficient equation</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq9_1.html" id="Eq9_1">
\begin{equation}
\frac{\mathrm{d}^2 U(x, s)}{\mathrm{d} x^2}-s U(x, s)=-3 \sin (2 \pi x).\tag{8.6.2}
\end{equation}
</div>
<p class="continuation">(b)The general solution of (<a href="" class="xref" data-knowl="./knowl/Eq9_1.html" title="Equation 8.6.2">(8.6.2)</a>) can be obtained by applying the method we have learned in ODE and we get</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq9_1.html">
\begin{equation*}
U(x, s)=C_1 e^{\sqrt{s} x}+C_2 e^{-\sqrt{s}x}+\frac{3}{(s+ 4\pi^2)} \sin (2 \pi x).
\end{equation*}
</div>
<p class="continuation">We note the Laplace transform of the boundary conditions give</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq9_1.html">
\begin{equation*}
u(0, t)=0 ~\to~ U(0, s)=0,\qquad u(2, t)=0 ~\to~ U(2, s)=0.
\end{equation*}
</div>
<p class="continuation">So we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq9_1.html">
\begin{equation*}
0=U(0, s)=C_1+C_2,\quad 0=U(2, s)=C_1 e^{2\sqrt{s} }+C_2 e^{-2\sqrt{s}}
\end{equation*}
</div>
<p class="continuation">which gives <span class="process-math">\(C_1=0, C_2=0\)</span> and we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq9_1.html">
\begin{equation*}
U(x, s)=\frac{3}{(s+4 \pi^2)} \sin (2 \pi x).
\end{equation*}
</div>
<p class="continuation">(c) From the table, we find that</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq9_1.html">
\begin{equation*}
{\mathcal L}^{-1}[U(x, s)]=3 e^{-4 \pi^2 t} \sin (2 \pi x).
\end{equation*}
</div>
<p class="continuation">Thus</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/Eq9_1.html">
\begin{equation*}
u(x, t)=3 e^{-4 \pi^2 t} \sin (2 \pi x).
\end{equation*}
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